Continuous Optimization Problems Research Articles

Trajectory optimization of spacecraft autonomous far-distance rapid rendezvous based on deep reinforcement learning

This paper investigates the application of Deep Reinforcement Learning (DRL) in the trajectory optimization of spacecraft far-distance rapid rendezvous, and uses the most advanced DRL method Proximal Policy Optimization (PPO) to solve the continuous high-thrust minimum-fuel trajectory optimization problem. The space J2 perturbation was considered, its impact on the spacecraft’s on-orbit operation and trajectory design was analyzed, and the effectiveness and accuracy of the proposed method were verified in two far-distance rapid rendezvous missions. In order to ensure the safety of the subsequent close-range operation phase, a safe area reward framework is proposed, and sparse and dense safe area reward functions are designed. The dense safe area reward function significantly improves the training efficiency of the algorithm on the basis of ensuring terminal performance. In addition, the modeling and analysis of possible uncertainties in the spacecraft’s orbit operation, including observation uncertainty, state uncertainty and control uncertainty, is carried out to verify the performance of the proposed method through simulation. For uncertainties, the closed-loop performance of the policy is also evaluated by performing Monte Carlo simulations. The results show that the PPO algorithm can effectively deal with the rendezvous problem in uncertainty environments. These preliminary results demonstrate the great potential of the DRL method in achieving autonomous guidance of spacecraft.

CMA-ES with Learning Rate Adaptation

The covariance matrix adaptation evolution strategy (CMA-ES) is one of the most successful methods for solving continuous black-box optimization problems. A practically useful aspect of CMA-ES is that it can be used without hyperparameter tuning. However, the hyperparameter settings still have a considerable impact on performance, especially for difficult tasks, such as solving multimodal or noisy problems. This study comprehensively explores the impact of learning rate on CMA-ES performance and demonstrates the necessity of a small learning rate by considering ordinary differential equations. Thereafter, it discusses the setting of an ideal learning rate. Based on these discussions, we develop a novel learning rate adaptation mechanism for CMA-ES that maintains a constant signal-to-noise ratio. Additionally, we investigate the behavior of CMA-ES with the proposed learning rate adaptation mechanism through numerical experiments and compare the results with those obtained for CMA-ES with a fixed learning rate and with population size adaptation. The results show that CMA-ES with the proposed learning rate adaptation works well for multimodal and/or noisy problems without extremely expensive learning rate tuning.

UAV Routing for Enhancing the Performance of a Classifier-in-the-loop

Some human-machine systems are designed so that machines (robots) gather and deliver data to remotely located operators (humans) through an interface to aid them in classification. The performance of a human as a (binary) classifier-in-the-loop is characterized by probabilities of correctly classifying objects (or points of interest) as a true target or a false target. These two probabilities depend on the time spent collecting information at a point of interest (POI), known as dwell time. The information gain associated with collecting information at a POI is then a function of dwell time and discounted by the revisit time, i.e., the duration between consecutive revisits to the same POI, to ensure that the vehicle covers all POIs in a timely manner. The objective of the routing problem for classification is to route the vehicles optimally, which is a discrete problem, and determine the optimal dwell time at each POI, which is a continuous optimization problem, to maximize the total discounted information gain while visiting every POI at least once. Due to the coupled discrete and continuous problem, which makes the problem hard to solve, we make a simplifying assumption that the information gain is discounted exponentially by the revisit time; this assumption enables one to decouple the problem of routing with the problem of determining optimal dwell time at each POI for a single vehicle problem. For the multi-vehicle problem, since the problem involves task partitioning between vehicles in addition to routing and dwell time computation, we provide a fast heuristic to obtain high-quality feasible solutions.

A tree seed algorithm with multi-strategy for parameter estimation of solar photovoltaic models

Tree seed algorithm, which is one of the metaheuristics algorithms recently proposed for the solution of continuous optimization problems, has an effective algorithmic structure inspired by the relation between trees and seeds. At the same time, the use of two different solution generation mechanisms by depending on the control parameter in TSA aims to balance the exploration and exploitation capabilities of the algorithm. However, when the structure of the algorithm is examined in detail, it is seen that there are some disadvantages such as loss of population diversity and getting stuck in local minimums. To overcome these disadvantages in the basic algorithm, three different approaches (self-adaptive weighting mechanism, chaotic elite learning approach and experience-based learning method) were proposed to TSA under the name of multi-strategies in this study. The algorithm improved with these approaches is named as the multi-strategy-based tree seed algorithm (MS-TSA). MS-TSA was first tested on CEC2017 functions. Then MS-TSA was applied to the problems in the CEC2020 competition and compared with the results of the best performing algorithms in this competition. As a result of the comparisons, MS-TSA was found to be a competitive method on solving benchmark functions. Then, parameter estimation of single diode, double diode and photovoltaic module models using the input data of various solar panels was carried out by the MS-TSA. The results obtained with MS-TSA were compared with both the results of the basic TSA and the results of well-known algorithms in the literature. The results obtained are 9.8642E-04, 9.8356E-04, 2.4251E-03, 1.7534E-03 respectively. As a result of the comparative analysis, the lowest RMSE value was obtained by MS-TSA. In addition, comprehensive performance analyzes of the algorithms were made with the convergence curve, boxplots, current (I) - voltage (V) and power (P) - voltage (V) characteristic curves obtained according to the experimental results. As a result of the experiments and analyses, MS-TSA was found to be a more successful method than the compared algorithms in parameter estimation of PV models.

Pflacco: Feature-Based Landscape Analysis of Continuous and Constrained Optimization Problems in Python.

The herein proposed Python package pflacco provides a set of numerical features to characterize single-objective continuous and constrained optimization problems. Thereby, pflacco addresses two major challenges in the area of optimization. Firstly, it provides the means to develop an understanding of a given problem instance, which is crucial for designing, selecting, or configuring optimization algorithms in general. Secondly, these numerical features can be utilized in the research streams of automated algorithm selection and configuration. While the majority of these landscape features are already available in the R package flacco, our Python implementation offers these tools to an even wider audience and thereby promotes research interests and novel avenues in the area of optimization.

JSOSuite.jl: Solving continuous optimization problems with JuliaSmoothOptimizers

JSOSuite.jl: Solving continuous optimization problems with JuliaSmoothOptimizers

AdaGuiDE: An adaptive and guided differential evolution for continuous optimization problems

Differential evolution (DE) has been proven as a simple yet powerful meta-heuristic algorithm on tackling continuous optimization problems. Nevertheless most existing DE methods still suffer from certain drawbacks including the use of ineffective mechanisms to adjust mutation strategies and their control parameters that may possibly mislead the search directions, and also the lack of intelligent guidance and reset mechanisms to escape from local optima. Therefore, to enhance the adaptability of DE-based search frameworks and the robustness on optimizing complex problems full of local optima, an adaptive and guided differential evolution (AdaGuiDE) algorithm is proposed. Essentially, the adaptability of the AdaGuiDE search framework is enhanced by three schemes to iteratively refine the search behaviour at two different levels. At the macroscopic level, the AdaGuiDE search framework revises the existing adaptive mechanism for selecting appropriate DE search strategies by counting the actual contributions in terms of solution quality. In addition, the adaption strategy is extended to the microscopic level where a penalty-based guided DE search is employed to guide the search escaping from local optima through temporarily penalizing the local optima and their neighborhood. Furthermore, a systematic boundary revision scheme is introduced to dynamically adjust the search boundary for locating any potential regions of interest during the search. For a rigorous evaluation of the proposed search framework, the AdaGuiDE algorithm is compared against other well-known meta-heuristic approaches on three sets of benchmark functions involving different dimensions in which the AdaGuiDE algorithm attained remarkable results especially on the high-dimensional and complex optimization problems. More importantly, the proposed AdaGuiDE framework shed lights on many possible directions to further enhance the adaptability of the underlying DE-based search strategies in tackling many challenging real-world applications.

An improved nutcracker optimization algorithm for discrete and continuous optimization problems: Design, comprehensive analysis, and engineering applications

This study is presented to examine the performance of a newly proposed metaheuristic algorithm within discrete and continuous search spaces. Therefore, the multithresholding image segmentation problem and parameter estimation problem of both the proton exchange membrane fuel cell (PEMFC) and photovoltaic (PV) models, which have different search spaces, are used to test and verify this algorithm. The traditional techniques could not find approximate solutions for those problems in a reasonable amount of time, so researchers have used metaheuristic algorithms to overcome those shortcomings. However, the majority of metaheuristic algorithms still suffer from slow convergence speed and stagnation into local minima problems, which makes them unsuitable for tackling these optimization problems. Therefore, this study proposes an improved nutcracker optimization algorithm (INOA) for better solving those problems in an acceptable amount of time. INOA is based on improving the performance of the standard algorithm using a newly proposed convergence improvement strategy that aims to improve the convergence speed and prevent stagnation in local minima. This algorithm is first applied to estimating the unknown parameters of the single-diode, double-diode, and triple-diode models for a PV module and a solar cell. Second, four PEMFC modules are used to further observe INOA's performance for the continuous optimization challenge. Finally, the performance of INOA is investigated for solving the multi-thresholding image segmentation problem to test its effectiveness in a discrete search space. Several test images with different threshold levels were used to validate its effectiveness, stability, and scalability. Comparison to several rival optimizers using various performance indicators, such as convergence curve, standard deviation, average fitness value, and Wilcoxon rank-sum test, demonstrates that INOA is an effective alternative for solving both discrete and continuous optimization problems. Quantitively, INOA could solve those problems better than the other rival optimizers, with improvement rates for final results ranging between 0.8355 % and 3.34 % for discrete problems and 4.97 % and 99.9 % for continuous problems.

Neural quantile optimization for edge–cloud networking

We seek the best traffic allocation scheme for the edge–cloud networking subject to SD-WAN architecture and burstable billing. First, we formulate a family of quantile-based integer programming problems for a fixed network topology with random parameters describing the traffic demands. Then, to overcome the difficulty caused by the discrete feature, we generalize the Gumbel-softmax reparameterization method to induce an unconstrained continuous optimization problem as a regularized continuation of the discrete problem. Finally, we introduce the Gumbel-softmax sampling neural network to solve optimization problems via unsupervised learning. The neural network structure reflects the edge–cloud networking topology and is trained to minimize the expectation of the cost function for unconstrained continuous optimization problems. The trained network works as an efficient traffic allocation scheme sampler, outperforming the random strategy in feasibility and cost value. Besides testing the quality of the output allocation scheme, we examine the generalization property of the network by increasing the time steps and the number of users. We also feed the solution to existing integer optimization solvers as initial conditions and verify the warm-starts can accelerate the short-time iteration process. The framework is general, and the decoupled feature of the random neural networks is adequate for practical implementations.

Improving quality of service for Internet of Things(IoT) in real life application: A novel adaptation based Hybrid Evolutionary Algorithm

In this paper, we address critical challenges in IoT sensor lifespan, service latency, and coverage area, all impacting energy consumption in smart agriculture applications. To enhance the quality of service (QoS) while prolonging the energy efficiency of smart sensors, a novel optimization algorithm is introduced. Referred to as the ”Adaptation-Based Hybrid Evolutionary Algorithm,” this innovative approach combines the strengths of Grey Wolf Optimizers (GWO) and Differential Evolution (DE) algorithms. The methodology involves a new adaptation-based strategy and incorporates a hybrid algorithm that synergizes the exploratory and exploitative capabilities of both GWO and DE algorithms. This hybrid approach is leveraged to meticulously select optimal mutation new adaptation services, drawing from the GWO and DE algorithm frameworks. Notably, the algorithm’s control parameters autonomously adjust through insights gained from prior evolutionary searches. Furthermore, we enhance the DE-based crossover technique by integrating the proficient search capabilities of the GWO algorithm, renowned for tackling continuous global optimization problems. To validate our approach, we apply it to IoT scenarios and optimize QoS through a fitness function that comprehensively accounts for energy consumption, coverage rate, lifespan, and latency. Comparative evaluations against standard algorithms underscore the superior performance of our proposed methodology, particularly evident in its application to IoT-smart agriculture settings.

FATA: An efficient optimization method based on geophysics

An efficient swarm intelligence algorithm is proposed to solve continuous multi-type optimization problems, named the fata morgana algorithm (FATA). By mimicking the process of mirage formation, FATA designs the mirage light filtering principle (MLF) and the light propagation strategy (LPS), respectively. The MLF strategy, combined with the definite integration principle, drives the algorithmic population to enhance FATA’s exploration capability. The LPS strategy, combined with the trigonometric principle, drives the algorithmic individual to improve the algorithm's convergence speed and exploitation capability. These two search strategies can better use FATA’s population and individual search capabilities. The FATA is compared with a broad array of competitive optimizers on 23 benchmark functions and IEEE CEC 2014 to verify the optimization capability. This work is designed separately for qualitative analysis, exploration and exploitation competence analysis, the analysis of avoiding locally optimal solutions, and comprehensive comparison experiments. The experimental results demonstrate the comprehensiveness and competitiveness of FATA for solving multi-type functions. Meanwhile, FATA is applied to three practical engineering optimization problems to evaluate its performance. Then, the algorithm obtains better results than its counterparts in engineering problems. According to the results, FATA has excellent potential to be used as an efficient computer-aided tool for dealing with practical optimization tasks. Source codes and related files are available at https://aliasgharheidari.com/FATA.html and other websites.

Metaheuristics for variable-size mixed optimization problems: A unified taxonomy and survey

Many real world optimization problems are formulated as mixed-variable optimization problems (MVOPs) which involve both continuous and discrete variables. MVOPs including dimensional variables are characterized by a variable-size search space. Depending on the values of dimensional variables, the number and type of the variables of the problem can vary dynamically. MVOPs and variable-size MVOPs (VMVOPs) are difficult to solve and raise a number of scientific challenges in the design of metaheuristics. Standard metaheuristics have been first designed to address continuous or discrete optimization problems, and are not able to tackle VMVOPs in an efficient way. The development of metaheuristics for solving such problems has attracted the attention of many researchers and is increasingly popular. However, to our knowledge there is no well established taxonomy or comprehensive survey for handling this important family of optimization problems. This paper presents an unified taxonomy for metaheuristic solutions for solving VMVOPs in an attempt to provide a common terminology and classification mechanisms. It provides a general mathematical formulation and concepts of VMVOPs, and identifies the various solving methodologies than can be applied in metaheuristics. The advantages, the weaknesses and the limitations of the presented methodologies are discussed. The proposed taxonomy also allows to identify some open research issues which needs further in-depth investigations.

Roulette wheel-based level learning evolutionary algorithm for feature selection of high-dimensional data

Feature selection in high-dimensional data is a large-scale sparse and discrete optimization problem. Most evolutionary algorithms are designed to tackle continuous optimization problems. However, when dealing with high-dimensional feature selection tasks, they often suffer from poor population diversity and are computationally expensive. To address these challenges, this work introduces a roulette wheel-based level learning evolutionary algorithm (RWLLEA). RWLLEA integrates two key components. Firstly, it employs a leveled population mode. Individuals from higher levels provide guidance to those at lower levels during the evolutionary process, thereby exploring the potential combinatorial effects among features. Secondly, recognizing the characteristics associated with the high-dimensional feature selection task, a roulette wheel-based update method is devised to dynamically reduce search space and harmonize the algorithm's exploitation and exploration capacities across different stages. The performance of the proposed method was evaluated by comparing it with six other feature selection techniques across a range of fifteen diverse datasets. The experimental findings demonstrate that the proposed method can achieve a reduced feature set with a shorter runtime and exhibit superior classification accuracy.

Advanced backtracking search for solving continuous optimization problems

Advanced backtracking search for solving continuous optimization problems

Opt2Vec - a continuous optimization problem representation based on the algorithm's behavior: A case study on problem classification

Characterization of the optimization problem is a crucial task in many recent optimization research topics (e.g., explainable algorithm performance assessment, and automated algorithm selection and configuration). The state-of-the-art approaches use exploratory landscape analysis to represent the optimization problem, where for each one, a set of features is extracted using a set of candidate solutions sampled by a sampling strategy over the whole decision space. This paper proposes a novel representation of continuous optimization problems by encoding the information found in the interaction between an algorithm and an optimization problem. The new problem representation is learned using the information from the states/positions in the optimization run trajectory (i.e., the candidate solutions visited by the algorithm). With the novel representation, the problem can be characterized dynamically during the optimization run, instead of using a set of candidate solutions from the whole decision space that have never been observed by the algorithm. The novel optimization problem representation is called Opt2Vec and uses an autoencoder type of neural network to encode the information found in the interaction between an optimization algorithm and optimization problem into an embedded subspace. The Opt2Vec representation efficiency is shown by enabling different optimization problems to be successfully identified using only the information obtained from the optimization run trajectory.

Optimized quantization parameter selection for video-based point cloud compression

Point clouds are sets of points used to visualize three-dimensional (3D) objects. Point clouds can be static or dynamic. Each point is characterized by its 3D geometry coordinates and attributes such as color. High-quality visualizations often require millions of points, resulting in large storage and transmission costs, especially for dynamic point clouds. To address this problem, the moving picture experts group has recently developed a compression standard for dynamic point clouds called video-based point cloud compression (V-PCC). The standard generates two-dimensional videos from the geometry and color information of the point cloud sequence. Each video is then compressed with a video coder, which converts each frame into frequency coefficients and quantizes them using a quantization parameter (QP). Traditionally, the QPs are severely constrained. For example, in the low-delay configuration of the V-PCC reference software, the quantization parameter values of all the frames in a group of pictures are set to be equal. We show that the rate-distortion performance can be improved by relaxing this constraint and treating the QP selection problem as a multi-variable constrained combinatorial optimization problem, where the variables are the QPs. To solve the optimization problem, we propose a variant of the differential evolution (DE) algorithm. Differential evolution is an evolutionary algorithm that has been successfully applied to various optimization problems. In DE, an initial population of randomly generated candidate solutions is iteratively improved. At each iteration, mutants are generated from the population. Crossover between a mutant and a parent produces offspring. If the performance of the offspring is better than that of the parent, the offspring replaces the parent. While DE was initially introduced for continuous unconstrained optimization problems, we adapt it for our constrained combinatorial optimization problem. Also, unlike standard DE, we apply individual mutation to each variable. Furthermore, we use a variable crossover rate to balance exploration and exploitation. Experimental results for the low-delay configuration of the V-PCC reference software show that our method can reduce the average bitrate by up to 43% compared to a method that uses the same QP values for all frames and selects them according to an interior point method.

Joint Dual Feature Distillation and Gradient Progressive Pruning for BERT compression

The increasing size of pre-trained language models has led to a growing interest in model compression. Pruning and distillation are the primary methods employed to compress these models. Existing pruning and distillation methods are effective in maintaining model accuracy and reducing its size. However, they come with limitations. For instance, pruning is often suboptimal and biased by transforming it into a continuous optimization problem. Distillation relies primarily on one-to-one layer mappings for knowledge transfer, which leads to underutilization of the rich knowledge in teacher. Therefore, we propose a method of joint pruning and distillation for automatic pruning of pre-trained language models. Specifically, we first propose Gradient Progressive Pruning (GPP), which achieves a smooth transition of indicator vector values from real to binary by progressively converging the values of unimportant units’ indicator vectors to zero before the end of the search phase. This effectively overcomes the limitations of traditional pruning methods while supporting compression with higher sparsity. In addition, we propose the Dual Feature Distillation (DFD). DFD adaptively globally fuses teacher features and locally fuses student features, and then uses the dual features of global teacher features and local student features for knowledge distillation. This realizes a “preview-review” mechanism that can better extract useful information from multi-level teacher information and transfer it to student. Comparative experiments on the GLUE benchmark dataset and ablation experiments indicate that our method outperforms other state-of-the-art methods.

Adaptive search space for stochastic opposition-based learning in differential evolution

Differential evolution (DE) is a practical evolutionary algorithm (EA) widely employed for addressing continuous optimization problems. Opposition-based learning (OBL) emerges as a potent method among the techniques enhancing EA performance. The BetaCOBL variant represents a pinnacle in this domain. However, BetaCOBL’s utilization of the promising regions of the search space remains partial, owing to its dependence on a non-adaptive framework. Consequently, its efficacy might dwindle as optimization progresses. We aimed to introduce an enhanced version of BetaCOBL, termed adaptive BetaCOBL (ABetaCOBL). ABetaCOBL commences by adapting the search space based on population distribution and subsequently identifying opposite solutions. We evaluated the efficacy of embedding ABetaCOBL into DE algorithms through experiments. Our experimental results substantiate that ABetaCOBL outperforms its precursor and resilient OBL variants (e.g., ABetaCOBL outperforms iBetaCOBL-eig in 19 out of 58 problems with NL-SHADE-LBC and in 22 out of 58 problems with NL-SHADE-RSP).

Binary metaheuristic algorithms for 0–1 knapsack problems: Performance analysis, hybrid variants, and real-world application

This paper examines the performance of three binary metaheuristic algorithms when applied to two distinct knapsack problems (0–1 knapsack problems (KP01) and multidimensional knapsack problems (MKP)). These binary algorithms are based on the classical mantis search algorithm (MSA), the classical quadratic interpolation optimization (QIO) method, and the well-known differential evolution (DE). Because these algorithms were designed for continuous optimization problems, they could not be used directly to solve binary knapsack problems. As a result, the V-shaped and S-shaped transfer functions are used to propose binary variants of these algorithms, such as binary differential evolution (BDE), binary quadratic interpolation optimization (BQIO), and binary mantis search algorithm (BMSA). These binary variants are evaluated using various high-dimensional KP01 examples and compared to several classical metaheuristic techniques to determine their efficacy. To enhance the performance of those binary algorithms, they are combined with repair operator 2 (RO2) to offer better hybrid variants, namely HMSA, HQIO, and HDE. Those hybrid algorithms are evaluated using several medium- and large-scale KP01 and MKP instances, as well as compared to other hybrid algorithms, to demonstrate their effectiveness. This comparison is conducted using three performance metrics: average fitness value, Friedman mean rank, and computational cost. The experimental findings demonstrate that HQIO is a strong alternative for solving KP01 and MKP. In addition, the proposed algorithms are applied to the Merkle-Hellman Knapsack Cryptosystem and the resource allocation problem in adaptive multimedia systems (AMS) to illustrate their effectiveness when applied to optimize those real applications. The experimental findings illustrate that the proposed HQIO is a strong alternative for handling various knapsack-based applications.

A Sampling Strategy for High-Dimensional, Simulation-Based Transportation Optimization Problems

When tackling high-dimensional, continuous simulation-based optimization (SO) problems, it is important to balance exploration and exploitation. Most past SO research focuses on the enhancement of exploitation techniques. The exploration technique of an SO algorithm is often defined as a general-purpose sampling distribution, such as the uniform distribution, which is inefficient at searching high-dimensional spaces. This work is motivated by the formulation of exploration techniques that are suitable for large-scale transportation network problems and high-dimensional optimization problems. We formulate a sampling mechanism that combines inverse cumulative distribution function sampling with problem-specific structural information of the underlying transportation problem. The proposed sampling distribution assigns greater sampling probability to points with better expected performance as defined by an analytical network model. Validation experiments on a toy network illustrate that the proposed sampling distribution has important commonalities with the underlying and typically unknown true sampling distribution of the simulator. We study a high-dimensional traffic signal control case study of Midtown Manhattan in New York City. The results show that the use of the proposed sampling mechanism as part of an SO framework can help to efficiently identify solutions with good performance. Using the analytical information for exploration, regardless of whether it is used for exploitation, outperforms benchmarks that do not use it, including standard Bayesian optimization. Using the analytical information for exploration only yields solutions with similar performance than when the information is used for exploitation only, reducing the total compute times by 65%. This paper sheds light on the importance of developing suitable exploration techniques to enhance both the scalability and the compute efficiency of general-purpose SO algorithms. Funding: T. Tay thanks the Agency for Science, Technology and Research (A*STAR) Singapore for funding his work. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2023.0110 .